analiza aiqbench
This research applies Fourier Neural Operators (FNOs) to solve the challenging problem of constructing Lyapunov functions for families of nonlinear dynamical systems. By treating the mapping from vector fields to Lyapunov functions as a learnable operator, the study enables data-driven stability analysis that generalizes across parameterized dynamics without retraining for each specific system. This approach bridges theoretical control theory with modern machine learning architectures to address fundamental issues in system stability.
Learning Lyapunov Operators for Nonlinear Systems
arXiv:2609.18894v1 Announce Type: cross Abstract: Constructing Lyapunov functions for nonlinear dynamical systems is a central problem in stability analysis, yet remains challenging. Lyapunov functions are commonly characterized as solutions to first-order partial differential equations (PDEs), but these solutions are typically obtained for single systems, limiting their reuse across systems. In this paper, we study the Lyapunov solution operator that maps a vector field to the corresponding Lyapunov function defined by a dissipation-based Lyapunov PDE. We establish that, on compact subsets of the domain of attraction and under exponential stability assumptions, this operator is well-defined, unique, and continuous with respect to perturbations of both the vector field and the dissipation function. These results provide a theoretical foundation for approximating Lyapunov functions uniformly over families of nonlinear systems. Building on these theoretical foundations, we employ Fourier Neural Operators (FNOs) as a data-driven approximation of the Lyapunov solution operator. Numerical experiments demonstrate that a single trained operator can accurately approximate the numerical Lyapunov functions across parameterized families of dynamics. This illustrates the potential of neural operators for approximating Lyapunov functions.
Abstract:Constructing Lyapunov functions for nonlinear dynamical systems is a central problem in stability analysis, yet remains challenging. Lyapunov functions are commonly characterized as solutions to first-order partial differential equations (PDEs), but these solutions are typically obtained for single systems, limiting their reuse across systems. In this paper, we study the Lyapunov solution operator that maps a vector field to the corresponding Lyapunov function defined by a dissipation-based Lyapunov PDE. We establish that, on compact subsets of the domain of attraction and under exponential stability assumptions, this operator is well-defined, unique, and continuous with respect to perturbations of both the vector field and the dissipation function. These results provide a theoretical foundation for approximating Lyapunov functions uniformly over families of nonlinear systems. Building on these theoretical foundations, we employ Fourier Neural Operators (FNOs) as a data-driven approximation of the Lyapunov solution operator. Numerical experiments demonstrate that a single trained operator can accurately approximate the numerical Lyapunov functions across parameterized families of dynamics. This illustrates the potential of neural operators for approximating Lyapunov functions.
Analysis of PDEs (math.AP); Machine Learning (cs.LG); Optimization and Control (math.OC)
arXiv:2609.18894 [math.AP]
(or arXiv:2609.18894v1 [math.AP] for this version)
https://doi.org/10.48550/arXiv.2609.18894 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Text integral sub licența deschisă a publicației · 274 cuvinte
Subiecte conexe
Time-delay Control Using a New Nonlinear Adaptive Law for Cable-Driven Robots
arXiv · 30 iul. 2026
Neural ODEs Meet Concurrent Learning: Stable Online Learning with Lyapunov Guarantees
arXiv · 30 sept. 2026
Universal Approximation of Nonlinear Operators and Their Derivatives
arXiv · 2 sept. 2026
Dynamical System Transfer Learning with Reduced Order Models
Towards Data Science · 5 sept. 2026